Practice identifying and writing equations for parabolas and hyperbolas using vertices, foci, directrices, and asymptotes.
Read each problem carefully. Show your work in the space provided, including any standard form, completed square, or key values you use.
Graphing, standard forms, foci, directrices, and asymptotes
Math - Grade 9-12
- 1
For the parabola (x - 2)^2 = 8(y + 1), find the vertex, focus, directrix, and direction of opening.
- 2
Write the equation of the parabola with vertex (-3, 4) and focus (-3, 1).
- 3
Rewrite y = x^2 - 6x + 11 in vertex form. Then state the vertex and direction of opening.
- 4
Write the equation of the parabola with focus (5, 2) and directrix x = 1.
- 5
For the equation y^2 + 4y = 12x - 8, put the parabola in standard form and find the vertex, focus, and directrix.
- 6
For the hyperbola (x - 1)^2/9 - (y + 2)^2/16 = 1, find the center, vertices, foci, and equations of the asymptotes.
- 7
Write the equation of the hyperbola with center (0, 0), vertices (0, 5) and (0, -5), and foci (0, 13) and (0, -13).
- 8
Find the equations of the asymptotes for the hyperbola (y - 3)^2/4 - (x + 1)^2/25 = 1.
- 9
Put 9x^2 - 16y^2 - 54x - 64y - 127 = 0 in standard form. Then state the center and transverse axis direction.
- 10
Classify x^2 - 4x - 8y + 20 = 0 as a parabola or hyperbola. Then write it in standard form and state its vertex.
- 11
A hyperbola has foci (-4, 0) and (4, 0). For every point on the hyperbola, the absolute difference of the distances to the foci is 6. Write the equation in standard form.
- 12
A parabolic reflector has cross-section y = (1/12)x^2 with vertex at the origin. Find the focus of the parabola.